Preface |
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xvii | |
Notation |
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xx | |
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1 | (47) |
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1.1 The Physics of Information |
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1 | (4) |
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1 | (1) |
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1.1.2 Concentration Behaviors |
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2 | (2) |
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4 | (1) |
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1.2 The Dimensionality of the Space |
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5 | (8) |
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1.2.1 Bandlimitation Filtering |
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5 | (2) |
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1.2.2 The Number of Degrees of Freedom |
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7 | (2) |
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9 | (2) |
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11 | (2) |
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1.3 Deterministic Information Measures |
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13 | (5) |
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14 | (1) |
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1.3.2 Kolmogorov Capacity |
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14 | (3) |
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1.3.3 Quantized Unit of Information |
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17 | (1) |
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1.4 Probabilistic Information Measures |
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18 | (15) |
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1.4.1 Statistical Entropy |
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19 | (2) |
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1.4.2 Differential Entropy |
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21 | (1) |
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22 | (1) |
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1.4.4 Quantized Typical Waveforms |
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23 | (2) |
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25 | (1) |
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26 | (3) |
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29 | (1) |
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1.4.8 Capacity with Gaussian Noise |
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30 | (3) |
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33 | (8) |
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1.5.1 The Low-Energy Regime |
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33 | (1) |
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1.5.2 The High-Energy Regime |
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34 | (1) |
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1.5.3 Quantized Radiation |
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35 | (2) |
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37 | (4) |
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41 | (2) |
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1.7 Summary and Further Reading |
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43 | (1) |
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1.8 Test Your Understanding |
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44 | (4) |
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48 | (39) |
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48 | (1) |
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49 | (6) |
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2.2.1 Bandlimited Signals |
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50 | (1) |
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2.2.2 Timelimited Signals |
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51 | (1) |
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2.2.3 Impossibility of Time-Frequency Limiting |
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52 | (1) |
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53 | (2) |
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2.3 Heisenberg's Uncertainty Principle |
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55 | (5) |
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2.3.1 The Uncertainty Principle for Signals |
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55 | (1) |
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2.3.2 The Uncertainty Principle in Quantum Mechanics |
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56 | (2) |
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2.3.3 Entropic Uncertainty Principle |
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58 | (1) |
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2.3.4 Uncertainty Principle Over Arbitrary Measurable Sets |
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58 | (1) |
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2.3.5 Converse to the Uncertainty Principle |
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59 | (1) |
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60 | (3) |
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2.4.1 Problems with the Folk Theorem |
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61 | (2) |
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2.5 Slepian's Concentration Problem |
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63 | (7) |
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65 | (1) |
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2.5.2 Most Concentrated Functions |
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66 | (2) |
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2.5.3 Geometric View of Concentration |
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68 | (2) |
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2.6 Spheroidal Wave Functions |
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70 | (5) |
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71 | (1) |
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2.6.2 The Helmholtz Equation |
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72 | (3) |
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2.7 Series Representations |
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75 | (6) |
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2.7.1 Prolate Spheroidal Orthogonal Representation |
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76 | (2) |
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2.7.2 Other Orthogonal Representations |
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78 | (1) |
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2.7.3 Minimum Energy Error |
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79 | (2) |
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2.8 Summary and Further Reading |
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81 | (2) |
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2.9 Test Your Understanding |
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83 | (4) |
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3 Functional Approximation |
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87 | (43) |
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3.1 Signals and Functional Spaces |
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87 | (1) |
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88 | (2) |
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3.3 Degrees of Freedom of Bandlimited Signals |
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90 | (5) |
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3.3.1 Computation of the N-Widths |
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91 | (4) |
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3.4 Hilbert--Schmidt Integral Operators |
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95 | (9) |
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3.4.1 Timelimiting and Bandlimiting Operators |
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98 | (2) |
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3.4.2 Hilbert--Schmidt Decomposition |
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100 | (2) |
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3.4.3 Singular Value Decomposition |
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102 | (2) |
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104 | (9) |
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3.5.1 Approximately Bandlimited Signals |
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104 | (1) |
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105 | (3) |
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3.5.3 Signals of Multiple Variables |
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108 | (3) |
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3.5.4 Hybrid Scaling Regimes |
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111 | (2) |
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113 | (5) |
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3.6.1 Robustness of Blind Sensing |
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115 | (1) |
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116 | (2) |
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118 | (5) |
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3.7.1 Robustness of Compressed Sensing |
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119 | (2) |
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3.7.2 Probabilistic Reconstruction |
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121 | (1) |
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3.7.3 Information Dimension |
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122 | (1) |
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3.8 Summary and Further Reading |
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123 | (1) |
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3.9 Test Your Understanding |
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124 | (6) |
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4 Electromagnetic Propagation |
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130 | (27) |
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130 | (2) |
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132 | (6) |
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4.2.1 Perfectly Conductive Media |
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134 | (3) |
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137 | (1) |
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4.3 Conservation of Power |
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138 | (1) |
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4.4 Plane Wave Propagation |
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139 | (5) |
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140 | (1) |
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141 | (1) |
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142 | (1) |
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143 | (1) |
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4.5 The Wave Equation for the Potentials |
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144 | (2) |
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146 | (6) |
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4.6.1 The Far-Field Region |
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149 | (2) |
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4.6.2 The Fraunhofer Region |
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151 | (1) |
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4.7 Equivalence and Uniqueness |
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152 | (1) |
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4.8 Summary and Further Reading |
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153 | (1) |
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4.9 Test Your Understanding |
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154 | (3) |
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5 Deterministic Representations |
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157 | (16) |
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157 | (2) |
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5.1.1 Four Field Representations |
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157 | (1) |
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5.1.2 The Space--Frequency Spectral Domain |
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158 | (1) |
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5.2 System Representations |
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159 | (5) |
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5.2.1 Linear, Time-Invariant Systems |
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159 | (1) |
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5.2.2 Linear, Time-Invariant, Homogeneous Media |
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160 | (1) |
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5.2.3 Green's Function in Free Space for the Potential |
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160 | (1) |
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5.2.4 Green's Function in Free Space for the Field |
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161 | (1) |
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5.2.5 Green's Function for Cylindrical Propagation |
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162 | (2) |
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5.3 Discrete Radiating Elements |
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164 | (3) |
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5.3.1 Single Transmitter--Receiver Pair |
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164 | (2) |
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5.3.2 Multiple Transmitters and Receivers |
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166 | (1) |
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5.3.3 Singular Value Decomposition |
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166 | (1) |
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5.4 Communication Systems: Arbitrary Radiating Elements |
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167 | (4) |
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5.4.1 Hilbert--Schmidt Decomposition |
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168 | (2) |
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5.4.2 Optimal Communication Architecture |
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170 | (1) |
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5.5 Summary and Further Reading |
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171 | (1) |
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5.6 Test Your Understanding |
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172 | (1) |
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6 Stochastic Representations |
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173 | (27) |
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173 | (1) |
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6.2 Green's Function for a Random Environment |
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174 | (2) |
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6.2.1 Linear, Time-Varying Systems |
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174 | (2) |
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6.2.2 Linear, Space-Time-Varying Systems |
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176 | (1) |
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176 | (10) |
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6.3.1 Frequency-Varying Green's Function: Coherence Bandwidth |
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179 | (2) |
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6.3.2 Time-Varying Green's Function: Coherence Time |
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181 | (2) |
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6.3.3 Mutual Coherence Function |
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183 | (3) |
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6.3.4 Spatially Varying Green's Function: Coherence Distance |
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186 | (1) |
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6.4 Karhunen--Loeve Representation |
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186 | (10) |
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6.4.1 Time-Varying Green's Function |
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187 | (3) |
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6.4.2 Optimality of the Karhunen--Loeve Representation |
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190 | (1) |
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6.4.3 Stochastic Diversity |
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191 | (2) |
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6.4.4 Constant Power Spectral Density |
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193 | (1) |
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6.4.5 Frequency-Varying Green's Function |
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194 | (1) |
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6.4.6 Spatially Varying Green's Function |
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195 | (1) |
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6.5 Summary and Further Reading |
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196 | (1) |
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6.6 Test Your Understanding |
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197 | (3) |
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7 Communication Technologies |
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200 | (30) |
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200 | (1) |
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200 | (5) |
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203 | (1) |
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204 | (1) |
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7.3 Overview of Current Technologies |
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205 | (3) |
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205 | (1) |
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205 | (1) |
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206 | (1) |
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206 | (1) |
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207 | (1) |
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7.4 Principles of Operation |
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208 | (11) |
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7.4.1 Orthogonal Spectrum Division |
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209 | (3) |
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7.4.2 Orthogonal Code Division |
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212 | (4) |
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7.4.3 Exploiting Diversity |
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216 | (1) |
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7.4.4 Orthogonal Spatial Division |
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217 | (2) |
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219 | (7) |
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219 | (2) |
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7.5.2 Hierarchical Cooperation |
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221 | (1) |
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7.5.3 Interference Alignment |
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222 | (2) |
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224 | (1) |
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225 | (1) |
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7.6 Summary and Further Reading |
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226 | (1) |
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7.7 Test Your Understanding |
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227 | (3) |
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8 The Space--Wavenumber Domain |
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230 | (35) |
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8.1 Spatial Configurations |
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230 | (1) |
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231 | (1) |
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8.3 The Field's Functional Space |
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232 | (1) |
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233 | (11) |
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8.4.1 Bandlimitation Error |
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234 | (2) |
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8.4.2 Phase Transition of the Bandlimitation Error |
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236 | (2) |
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8.4.3 Asymptotic Evaluation |
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238 | (2) |
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240 | (3) |
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8.4.5 Size of the Transition Window |
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243 | (1) |
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244 | (6) |
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8.5.1 Hilbert--Schmidt Decomposition |
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246 | (2) |
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248 | (2) |
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250 | (8) |
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8.6.1 Linear Cut-Set Integral |
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251 | (2) |
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8.6.2 Surface Cut-Set Integral |
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253 | (3) |
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8.6.3 Applications to Canonical Geometries |
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256 | (2) |
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258 | (3) |
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8.8 Summary and Further Reading |
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261 | (1) |
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8.9 Test Your Understanding |
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261 | (4) |
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9 The Time--Frequency Domain |
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265 | (10) |
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9.1 Frequency-Bandlimited Signals |
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265 | (1) |
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9.2 Radiation with Arbitrary Multiple Scattering |
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266 | (6) |
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9.2.1 Two-Dimensional Circular Domains |
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267 | (2) |
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9.2.2 Three-Dimensional Spherical Domains |
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269 | (1) |
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9.2.3 General Rotationally Symmetric Domains |
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270 | (2) |
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272 | (1) |
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9.4 Alternative Derivations |
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273 | (1) |
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9.5 Summary and Further Reading |
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274 | (1) |
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9.6 Test Your Understanding |
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274 | (1) |
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10 Multiple Scattering Theory |
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275 | (28) |
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10.1 Radiation with Multiple Scattering |
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275 | (3) |
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10.1.1 The Basic Equation |
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276 | (1) |
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10.1.2 Multi-path Propagation |
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277 | (1) |
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10.2 Multiple Scattering in Random Media |
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278 | (6) |
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10.2.1 Born Approximation |
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281 | (1) |
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10.2.2 Complete Solutions |
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281 | (2) |
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283 | (1) |
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284 | (7) |
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10.3.1 Radiated Power Density |
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289 | (1) |
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10.3.2 Full Power Density |
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289 | (1) |
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290 | (1) |
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290 | (1) |
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10.4 Path Loss Measurements |
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291 | (1) |
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10.5 Pulse Propagation in Random Media |
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292 | (7) |
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10.5.1 Expected Space-Time Power Response |
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292 | (4) |
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10.5.2 Random Walk Interpretation |
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296 | (1) |
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10.5.3 Expected Space--Frequency Power Response |
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297 | (1) |
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10.5.4 Correlation Functions |
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298 | (1) |
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10.6 Power Delay Profile Measurements |
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299 | (1) |
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10.7 Summary and Further Reading |
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300 | (1) |
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10.8 Test Your Understanding |
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301 | (2) |
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303 | (40) |
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11.1 Measurement Uncertainty |
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303 | (4) |
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303 | (1) |
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304 | (2) |
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306 | (1) |
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306 | (1) |
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307 | (7) |
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11.2.1 Radiation Law, Classical Derivation |
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307 | (4) |
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11.2.2 Thermal Noise, Classical Derivation |
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311 | (1) |
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11.2.3 Quantum Mechanical Correction |
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312 | (2) |
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11.3 Equilibrium Configurations |
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314 | (8) |
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11.3.1 Statistical Entropy |
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316 | (1) |
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11.3.2 Thermodynamic Entropy |
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317 | (1) |
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11.3.3 The Second Law of Thermodynamics |
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318 | (1) |
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11.3.4 Probabilistic Interpretation |
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319 | (1) |
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11.3.5 Asymptotic Equipartition Property |
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320 | (1) |
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321 | (1) |
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322 | (1) |
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11.5 The Microwave Window |
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323 | (2) |
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11.6 Quantum Complementarity |
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325 | (1) |
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11.7 Entropy of a Black Body |
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326 | (3) |
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326 | (1) |
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11.7.2 Thermodynamic Entropy |
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327 | (1) |
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328 | (1) |
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11.7.4 Gravitational Limits |
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329 | (1) |
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11.8 Entropy of Arbitrary Systems |
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329 | (2) |
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11.8.1 The Holographic Bound |
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330 | (1) |
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11.8.2 The Universal Entropy Bound |
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330 | (1) |
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11.9 Entropy of Black Holes |
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331 | (2) |
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11.10 Maximum Entropy Distributions |
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333 | (3) |
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11.11 Summary and Further Reading |
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336 | (2) |
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11.12 Test Your Understanding |
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338 | (5) |
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12 Information-Theoretic Quantities |
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343 | (48) |
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12.1 Communication Using Signals |
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343 | (1) |
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344 | (14) |
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347 | (2) |
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349 | (1) |
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12.2.3 Capacity and Mutual Information |
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350 | (2) |
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352 | (2) |
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12.2.5 Quantum Constraints |
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354 | (1) |
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12.2.6 Capacity of the Noiseless Photon Channel |
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355 | (1) |
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12.2.7 Colored Gaussian Noise |
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356 | (1) |
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12.2.8 Minimum Energy Transmission |
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357 | (1) |
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12.3 A More Rigorous Formulation |
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358 | (6) |
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12.3.1 Timelimited Signals |
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359 | (1) |
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12.3.2 Bandlimited Signals |
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360 | (2) |
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12.3.3 Refined Noise Models |
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362 | (2) |
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364 | (4) |
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12.4.1 Rate-Distortion Function |
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364 | (2) |
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12.4.2 Rate-Distortion and Mutual Information |
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366 | (2) |
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12.5 Kolmogorov's Deterministic Quantities |
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368 | (3) |
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12.5.1 e-Coverings, e-Nets, and e-Entropy |
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369 | (1) |
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12.5.2 e-Distinguishable Sets and e-Capacity |
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370 | (1) |
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12.5.3 Relation Between e-Entropy and e-Capacity |
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370 | (1) |
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12.6 Basic Deterministic--Stochastic Model Relations |
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371 | (3) |
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371 | (2) |
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373 | (1) |
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12.7 Information Dimensionality |
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374 | (4) |
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374 | (2) |
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12.7.2 Functional Dimension and Metric Order |
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376 | (1) |
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12.7.3 Infinite-Dimensional Spaces |
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377 | (1) |
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378 | (6) |
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12.8.1 Capacity and Packing |
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378 | (1) |
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12.8.2 Entropy and Covering |
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378 | (1) |
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12.8.3 e-Capacity of Bandlimited Signals |
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379 | (2) |
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12.8.4 (e, δ)-Capacity of Bandlimited Signals |
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381 | (1) |
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12.8.5 e-Entropy of Bandlimited Signals |
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382 | (1) |
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12.8.6 Comparison with Stochastic Quantities |
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383 | (1) |
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12.9 Spatially Distributed Systems |
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384 | (4) |
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12.9.1 Capacity with Channel State Information |
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384 | (3) |
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12.9.2 Capacity without Channel State Information |
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387 | (1) |
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12.10 Summary and Further Reading |
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388 | (1) |
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12.11 Test Your Understanding |
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389 | (2) |
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13 Universal Entropy Bounds |
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391 | (16) |
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13.1 Bandlimited Radiation |
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391 | (1) |
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13.2 Deterministic Signals |
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392 | (5) |
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13.2.1 Quantization Error |
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393 | (1) |
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13.2.2 Kolmogorov Entropy Bound |
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394 | (2) |
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13.2.3 Saturating the Bound |
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396 | (1) |
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397 | (3) |
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13.3.1 Shannon Rate--Distortion Bound |
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398 | (1) |
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13.3.2 Shannon Entropy Bound |
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398 | (2) |
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13.4 One-Dimensional Radiation |
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400 | (1) |
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401 | (2) |
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13.5.1 High-Energy Limits |
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401 | (1) |
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13.5.2 Relation to Current Technologies |
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402 | (1) |
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13.6 On Models and Reality |
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403 | (2) |
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13.7 Summary and Further Reading |
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405 | (1) |
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13.8 Test Your Understanding |
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406 | (1) |
Appendix A Elements of Functional Analysis |
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407 | (15) |
Appendix B Vector Calculus |
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422 | (6) |
Appendix C Methods for Asymptotic Evaluation of Integrals |
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428 | (5) |
Appendix D Stochastic Integration |
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433 | (1) |
Appendix E Special Functions |
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434 | (3) |
Appendix F Electromagnetic Spectrum |
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437 | (1) |
Bibliography |
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438 | (9) |
Index |
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447 | |